The theorem of Paul Hall [1]} states that the group algebra of a virtually polycyclic group is Noetherian.
Recall that a finitely generated group G is polycyclic if it admits a subnormal series whose factors are all cyclic groups.
Note that nilpotent groups are polycyclic, polycyclic groups are solvable and solvable groups are amenable.
The lamplighter group is an example of a solvable group which is not polycyclic.
Since finite extensions of amenable groups are amenable,
it follows that virtually polycyclic groups are amenable.
Thanks to P. Hall's theorem, we get a large family of groups to which the above results apply.
